Abstract algebra by Choudhary P.

By Choudhary P.

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As ???????????? ∘ ????????????,???? : ????????,???? → ????????′ = lim???? ′ ∈???? ????????′ ,???? ′ and ????????,???? is finitely presented, there are ???? ′ ∈ ????????′ and −→ ????′ ???? : (????, ????) → (????′ , ???? ′ ) in ????. Hence, ′ ′ ????????′ ∘ ???????????? ∘ ????????????,???? = ????????′ ∘ ????????????′ ,???? ′ ∘ ???? = ????????′ ,???? ′ ∘ ???? = ????????,???? = ???????? ∘ ????????????,???? . ′ As this holds for all ???? ∈ ???????? , it follows that ????????′ ∘ ???????????? = ???????? . Hence, as ???? = lim????∈???? ???????? , there exists a unique morphism ???? : ???? → ???? such that ???????? = ???? ∘ ???????? −→ for each ???? ∈ ????. It follows that ???? ∘ ????????,???? = ???? ∘ ???????? ∘ ????????????,???? = ???????? ∘ ????????????,???? = ????????,???? , for all (????, ????) ∈ ????.

Hence ????)= ???????? (????), ⋁ (????) (????) (????) ???????? ∣ ???? ≥ ????, ???? , and so where ???? := ????′ ∧ ????′′ belongs to ???????? ∩ ???????? = ⋁ ???? = (???????? ∣ ???? ∈ ????) in ???????? , for some finite set ???? of upper bounds of {????, ????} and ⋁ (????) elements )) ⋁ ( (????)???????? ∈ ???????? , )for all ???? ∈ ????. Therefore, ???? = (???????? (????(???? ) ∣ (???? ∈(????)????) belongs to ???? ∣ ???? ≥ ????, ???? . This completes the proof that ???? = ????, ???? ∣ ???? ∈ ???? is a ???? -scaled Boolean algebra. The verification of the statement ( ) (????, ???????? ∣ ???? ∈ ????) = lim ???????? , ???????????? ∣ ???? ≤ ???? in ???? −→ is routine.

We put ′ ′ ????????????,???? := ???????????? ∘ ????????????,???? , for all ???? ≤ ????′ in ???? and all ???? ∈ ???????? . Claim. The category ???? is filtered. Proof. First let (????0 , ????0 ), (????1 , ????1 ) ∈ ???? . There exists ???? ≥ ????0 , ????1 in ????. As ???????????? ,???????? is finitely presented, there exists ????????′ ∈ ???????? such that ???????????????? ,???????? factors through ????????,????????′ , for all ???? < 2. Hence, taking ???? ≥ ????0′ , ????1′ in ???????? , both morphisms ????????????0 ,????0 and ????????????1 ,????1 factor through ????????,???? , which yields ???????? : (???????? , ???????? ) → (????, ????), for all ???? < 2.

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